Exact evaluation of the cutting path length in a percolation model on a hierarchical network

Abstract

This work presents an approach to evaluate the exact value of the fractal dimension of the cutting path dCPf on hierarchical structures with finite order of ramification. Our approach is based on a renormalization group treatment of the universality class of watersheds. By making use of the self-similar property, we show that dCPf depends only on the average cutting path (CP) of the first generation of the structure. For the simplest Wheastone hierarchical lattice (WHL), we present a mathematical proof. For a larger WHL structure, the exact value of dCPf is derived based on a computer algorithm that identifies the length of all possible CP's of the first generation.